Which Grasshopper Wins the Race?

Date: 06/11/2003 at 11:43:56
From: Kathy
Subject: Winning distance

Two grasshoppers have a race. The course is 12 feet long one way, and 
they race down and back for a total of 24 feet. They both start at the 
signal. The larger grasshopper covers 10 inches per jump, while the 
smaller one covers 5 inches. However, the smaller grasshopper takes 2 
jumps for every jump of the larger grasshopper. One of them manages to 
win. Which one and why?

Date: 06/11/2003 at 13:56:02
From: Doctor Ian
Subject: Re: Winning distance

Hi Kathy,

One thing that makes this confusing is that the jumps are measured in
inches, while the course distance is measured in feet. Let's measure
everything in inches:
                                           |
  A                                        |
   ----------------------------------------|  Turn around here
  B             144 inches                 |
                                           |

Now, what's interesting here is that the course length isn't a
multiple of 5 or 10, so it won't be an even number of hops from one
end to the middle. 

Let's say they've hopped to the 140-inch line. Since they each jump
10 inches in the same amount of time, they get there together. 

                   |
       A           |  
    ...|--|--|--|--|
       B           |
                   |

But there are only four inches to the turnaround point. The smaller
grasshopper will jump once, turn around, and jump again:

                   |
        <--------------  second jump
       A-------------->  first jump
    ...|--|--|--|--|..|
       B           |
                   |

In the same amount of time, the larger grasshopper will jump once:

        <-------------- 
       A-------------->         
    ...|--|--|--|--|..|..|..|..|..|..|
       B----------------------------->           
                   |

So now the smaller one is in the lead. So the question is, will he
keep the lead all the way back to the finish line? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/